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Transformations

Transformations


A transformation is a mathematical function that repositions points in a one-dimensional, two-dimensional, three-dimensional, or any other n -dimensional space. In this article, only transformations in the familiar twodimensional rectangular coordinate plane will be discussed.

Transformations map one set of points onto another set of points, generally with the purpose of changing the position, size, and/or shape of the figure made up by the first set of points. The first set of points, from the domain of the transformation, is called the set of pre-images, whereas the second set of points, from the range of the transformation, is called the set of images. Therefore, a transformation maps each pre-image point to its image point.

Reflections

Reflections are transformations that involve "flipping" points over a given line; hence, this type of transformation is sometimes called a "flip." When a figure is reflected in a line, the points on the figure are mapped onto the points on the other side of the line which form the figure's mirror image.

For example, in the first figure below, the triangle ABC, the pre-image figure, is reflected in the x -axis to produce the image triangle ABC. Note that if triangle ABC is traversed from A to B to C and back to A, the direction of this movement is counterclockwise. If triangle ABC is traversed from A to B to C and back to A, the direction is clockwise. This "reversal of orientation" is similar to the way images in a mirror are reversed, and is a fundamental property of all reflections.

In the case of the reflection in the x -axis, as seen in the first figure, the first coordinate of each image point is the same as the first coordinate of its pre-image point, but the second coordinate of any image point is the opposite, or negative, of the second coordinate of its pre-image point. Mathematically, we say that for a reflection in the x -axis, pre-image points of the form (x, y ) are mapped to image points of the form (x, y ), or, more compactly, r (x, y ) = (x, y ), where r represents the reflection.

Such a formula is sometimes called an image formula, since it shows how a transformation acts on a pre-image point to produce its image point. A reflection in the y -axis leaves all second coordinates the same but replaces each first coordinate with its opposite; therefore, an image formula for a reflection in the y -axis may be written r (x, y ) = (x, y ). The image formula r (x, y ) = (y, x ) represents a reflection in the line y = x. When this reflection is done, the coordinates of each pre-image point are reversed to give the image.

Rotations

A second type of transformation is the rotation, also known as a "turn." A rotation, as its name suggests, takes pre-image points and rotates them by some specified angle measure about some fixed point. In the figure below, the pre-image triangle CDE has been rotated 90° about the origin of the coordinate system to produce the image triangle CDE.

The image formula for a 90° rotation is R (x, y ) = (y, x ). It can be shown that, in general, the image formula for a rotation of angle measure t about the origin has image formula R (x, y ) = [x cos(t ) y sin(t )], [x sin(t ) + y cos(t )]. If t is positive, the direction of the rotation is counterclockwise; if t is negative, then the rotation is clockwise.

Translations

Another type of transformation is the translation or "slide." Translations take pre-image points and move them a given number of units horizontally and/or a given number of units vertically. A translation image formula has the form T (x, y ) = (x + a, y + b ), where a is the number of units moved horizontally and b is the number of units moved vertically. If a is positive, the horizontal shift is to the right. If a is negative, then it is to the left. Similarly, if b is positive, the vertical shift is upward; but if b is negative, the vertical shift is downward.

In the figure below, the pre-image triangle CDE has been translated 4 units to the left and 2 units upward to give the image triangle CDE. The image formula would be T (x, y ) = (x + (4), y + 2) = (x 4, y + 2).

Reflections, rotations, and translations change only the location of a figure. They have no effect on the size of the figure or on the distance between points in the figure. For this reason they are called "isometries" from the Greek words meaning "same measure." An isometry is also known as a "distance-preserving transformation." The next transformation discussedthe dilationis not an isometry; that is, it does not preserve distance.

Dilations

A dilation is also known as a "stretch" or "shrink" depending on whether the image figure is larger or smaller than the pre-image figure. The image formula for a dilation is d(x, y ) = (kx, ky ), where k is a real number, called the magnitude or scale factor, and where the center of the dilation is the origin. If k > 1, the image of the dilation is an enlargement of the pre-image on the same side of the center as the pre-image. Part (a) of the boxed figure illustrates this for k = 3.

If 0 < k < 1, the image of the dilation is a reduction in size of the pre-image on the same side of the center as the pre-image. Part (b) of the figure illustrates this for k = ½.

If k < 1, the image of the dilation is an enlargement of the pre-image on the opposite side of the center from the pre-image. Part (c) of the figure illustrates this for k = 3.

If 1 < k < 0, the image of the dilation is a reduction in size of the pre-image on the opposite side of the center from the pre-image. Part (d) of the figure illustrates this for k = ½.

Notice that the effect of a negative value for k is equivalent to that of a dilation with a magnitude equal to the absolute value of k followed by a rotation of 180° about the center of the dilation.

All of the transformations discussed above can easily be extended into 3-dimensional space. For example, a dilation in the 3-dimensional xyz -coordinate system would have an image formula of the form d(x, y, z ) = (kx, ky, kz ). When transformationsreflections, rotations, translations, and dilationsare expressed as matrices (as is taught in linear algebra), they can then be combined to create the movement of figures in computer animation programs.

see also Mapping, Mathematical; Tessellations.

Stephen Robinson

Bibliography

Serra, Michael. Discovering Geometry. Emeryville, CA: Key Curriculum Press, 1997.

Narins, Brigham, ed. World of Mathematics, Detroit: Gale Group, 2001.

Usiskin, Zalman, Arthur Coxford, Daniel Hirschhorn, Virginia Highstone, Hester Lewellen, Nicholas Oppong, Richard DiBianca, and Merilee Maeir. Geometry. Glenview, IL: Scott Foresman and Company, 1997.

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Transformations

TRANSFORMATIONS

Wilfred R. Bion conceived of transformations as the changes that the analysand's sense impressions of emotional experience undergo to become a progressive series of mental realizations. They are akin to the transformations that food undergoes in the digestive system to become protoplasm. The concept also belongs to the mathematical concepts that Bion employed to bring more rigor to psychoanalytic understanding. In this way, he was attempting to make psychoanalysis more scientific. This period in his work overlaps with the next period in his work, developing psychoanalysis as an "intuitionistic science."

Briefly put, Bion envisions psychoanalytic transformations as the psychoanalyst's attempt to help the analysand transform that part of an emotional experience of which he is unconscious into an emotional experience of which he is conscious. Transforming here would be changing the form but not the fundamental nature or invariant aspect of the emotional experience. In this way the analyst helps the analysand achieve private or personal knowledge about his emotional life. Bion states his theory in a rigorous mathematical notation in which his symbols have the following meanings: O = the symptom or analytic object; T = transformation; t = the representation of the transformation; K = knowledge link; β = beta elements, sense impressions of emotional experience; α = alpha elements, elements suitable for further mental processing; p = patient; a = analyst. Bion states, "I shall regard only those aspects of the patient's behavior which are significant as representing his view of O ; I shall understand what he says or does as if it were an artist's painting. In the session the facts of his behavior are like facts of a painting and from them I must find the nature of his representations (or, in terms of my notation, the nature of that which I denote by the sign T (patient)β). From the analytic treatment as a whole I hope to discover from the invariants in this material what O is, what he does to transform O (that is to say, the nature of T (patient)α) and, consequently, the nature of T (patient). This last point is the set of transformations in the group of transformations, to which his particular transformation (T (patient)) is to be assigned. As I am concerned with the nature (or . . . meaning) of these phenomena, my problem is to determine the relationship between the unknowns: T (patient), T (patient)α, and T (patient)β. Only in the last of these have I any facts on which to work. . . . I shall make three assumptions: (i) that the patient is talking about something (O ); (ii) that something, O, has impressed him and that he has transformed the impression by the process represented by Tp α and (iii) that his representation tp β is comprehensible" (Bion, 1965).

Bion considers the emotional experience of the analysand and of the analyst to be O (the symptom or analytic object) but each has his distinct experience of O : Op (patient), and Oa (analyst). The analyst must, with his alpha function, deduce the transformation O Tp α Tp β, which is then translated by the analyst as O Ta α Ta β.

Bion considers there to be four kinds of transformations in clinical practice: (a) "rigid-motion transformations," which involve little alteration and which correspond directly to past events that may now be relived in the (classical) transference; (b) "projective transformations," which correspond to Melanie Klein's concept of projective identification; (c) "transformations in hallucinosis," which occur only in psychosis, and (d) "transformation in O," by which Bion seems to mean a transformation both from the ineffable nature of the analytic object, the analysand's symptom, and through K (the knowledge link), to yet another state, that of Absolute Truth or Ultimate Reality, the Godhead.

Bion states, "The bearing on psycho-analysis and interpretation of what I have said may seem obscure; it is this: The beginning of a session has the configuration already formulated in the concept of the God-head. From this there evolves a pattern and at the same time the analyst seeks to establish contact with the evolving pattern. This is subject to his Transformation and culminates in his interpretation Ta b. I am aware of the problems I have left without attempting an approach to their solution. . . . In this book I draw attention to a few of the problems which present themselves in analytic material and offer suggestions for clarifying, first observation and then, assessment of what has been observed" (1965).

Transformations (1965) seems to represent Bion's last venture in employing mathematical notation to bring scientific rigor to clinical psychoanalytic phenomena. Transformations in O constitute "a bridge to a new science," the intuitionistic, to which he thereafter bent his efforts. Though mathematical, Transformations was the third in a series of foundational works that was gradually to alter how analysts regarded clinical material and their personal (T (analyst)) relationship to it. Having already defined the mind that had to develop in order to think "the thoughts without a thinker" (beta elements, emotional experiences in themselves), he then undertook to define how these thoughts evolve from sense impressions of emotional experience (beta elements) to alpha elements suitable for further mental processing. From there he defined the steps of "mentally digestive transformation" that these beta, and then alpha, elements must undergo in the analysand and in the analyst in order to qualify for status in a scientific deductive system or fall by the wayside because of invidious K. O is the beginning and the end of the transformational cycle. Bion's epistemological transformation of psychoanalytic metapsychology was then in place.

James S. Grotstein

See also: Catastrophic change; Dream-like memory; Grid; Hallucinosis; Invariant; Love-Hate-Knowledge (L/H/K links); Vertex.

Bibliography

Bion, Wilfred R. (1965). Transformations: Change from learning to growth. London: Heinemann.

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